Low volume bookshelf spkrs that go from 20 to 20k?

There are no bookshelf speakers on earth that can go to 20HZ in fact many extremely expensive speakers can't do it.

You will not need a subwoofer for a 14 foot square room. The small size and square shape will leed to severe standing waves and bass bloat.

If possible try to find a narrow floorstanding speaker system.

Resolution is the key to low volume. Resolution is dependent on the electronics. Try to find tube electronics which excel at low volumes.

Look at Dali loudspeakers which offer exceptional transparency and yet are warm and full bodied.

Two speaker makes come to mind which specifically seem to address the issue of constant sound quality acorss the volume spectrum - ie, they sound good at low, med or high volumes. Revel and Joseph audio - both make monitors that can be had in this price range. Revel especially.

Most speakers in my opinion do not maintain a consistent freq response curve at volume extremes and many sound lifeless at low volumes.

I would also say the a true bookshelf speaker is not going tp go down to 20hz - and if it came close it would vibrate itself and everything else right off the shelf! So assume you will need a subwoofer.

But if you are living in India (I'm guessing?) there are plenty of UK, French and Hong Kong products at 220v that might be easier to acquire than US brands.

Keep one other issue in mind: Your daughter's hearing is way better than the average middle aged male. What seems detailed in the high end to you will probably sound harsh to her? I would be careful on the electronics for that reason also.

Finally, kids love bass. They don't notice if it rolls off at 35hz. They just want to make it thump sometimes. Find her some fun looking gear?

Following up on what Narrod offered, a square room that size will make the bass response very uneven because there will be a large bass boost at some bass frequency. This will make the bass way out of true proportion whenever you hit notes in the boosted frequency band.

So trying for "full-range" in such a room could be a frustrating and futile experience. And the more powerful the speaker's bass response is, the more this uneven response will be exascerbated.

On the other hand, if you can calculate or measure the frequency of the bass hump, you may be able to find monitor speakers that roll-off before that point so that the big bump corresponds to the speaker's natural roll off.

Anyway, if you plan on going deep in the bass in that room you will probably need some judicious room equalization.

But just so we are on the same page, you do realize that most people can't actually hear a 20Hz tone but you can tell it's there because it will physically vibrate/shake the room and other objects within the room... There are almost no instruments that play down that low except for organ.

I see a Proac 1SC for sale here I would love to own.. not really bookshelf (requires stands, not against a wall) but would work great and very high quality for $1K. The only way to go down to 20Hz would be to have a pair of Allisons or AR's or other air suspension design used with the Allison 'Electronic Subwoofer' equalizer (way, way no longer available and almost never seen for sale).